Numerical algorithms for cold-relativistic plasma models in the presence of discontinuties

A numerical algorithm is presented to solve cold-relativistic electron fluid equations in the presence of sharp gradients and discontinuities. The intended application is to laser wake-field accelerator simulations in which the laser induces accelerating fields thousands of times those achievable in conventional RF accelerators. The relativistic cold-fluid equations are formulated as non-classical system of hyperbolic balance laws. It is shown that the flux Jacobian for this system can not be diagonalized which causes numerical difficulties when developing shock-capturing algorithms. Further, the system is shown to admit generalized delta-shock solutions, first discovered in the context of sticky-particle dynamics (Bouchut, \emph{Ser. Adv. Math App. Sci.}, {\bf 22} (1994) pp. 171--190). A new approach, based on relaxation schemes proposed by Jin and Xin (\emph{Comm. Pure Appl. Math.} {\bf 48} (1995) pp. 235--276) and LeVeque and Pelanti (\emph{J. Comput. Phys.} {\bf 172} (2001) pp. 572--591) is developed to solve this system of equations. The method consists of finding an exact solution to a Riemann problem at each cell interface and coupling these to advance the solution in time. Applications to an intense laser propagating in an under-dense plasma are presented.